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HMMT 十一月 2010 · GEN1 赛 · 第 7 题

HMMT November 2010 — GEN1 Round — Problem 7

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 6 ] What is the remainder when (1 + x ) is divided by 1 + x + x ?
解析
  1. [ 6 ] What is the remainder when (1 + x ) is divided by 1 + x + x ? 2 Answer: 1 We use polynomial congruence mod 1 + x + x to find the desired remainder. Since 2 3 3 2 x + x + 1 | x − 1, we have that x ≡ 1 (mod 1 + x + x ). Now: 2010 2 2010 2 (1 + x ) ≡ ( − x ) (mod 1 + x + x ) 4020 2 ≡ x (mod 1 + x + x ) 3 1340 2 ≡ ( x ) (mod 1 + x + x ) 1340 2 ≡ 1 (mod 1 + x + x ) 2 ≡ 1 (mod 1 + x + x ) Thus, the answer is 1.